The eye is a tightly packed character clue – in Poe that's about all we get, clues. The eye casts its gaze on us, through the story, and we are torn between fear and embrace. It's the only description of personal appearance in the story. Thanks to the Internet, we can offer a diagnoses – the narrator has a corneal hernia or some similar condition. Untreated this can result in all sorts of big problems. The old man's seems to be going untreated. This suggests that eye characterizes him as ill, and possibly neglected. In turn, this characterizes the narrator and the outlying community.
Or, maybe the eye is a symbol of the man's inner vision, his view of himself, his view of the world. His sight is veiled, yet piercing in a threatening way. From the narrator's perspective, we are never quite sure what the eye is or what the eye does. We do know that the eye is represented as almost an entity separate and apart from the old man. Yet, the narrator can't approach it directly, but must go through the man. The eye also seems to be somehow connected to the heart – the eye sees the tale, but the heart tells it.
Watching someone while they sleep for seven nights in a row constitutes a habit. The narrator has maybe done this sort of thing before. This characterizes him as somebody who you don't invite to functions where sleeping is involved. Unless he gets help.
The narrator's spying, plotting, and murdering characterize him as a dangerous person. His double confession suggests he has a conscience. The old man's lack of action in the face of danger characterizes him as weak and helpless.